Every form of matter, regardless of size or scale, ranging from atoms and molecules to columns and skyscrapers, is subjected to physical interaction with the environment. These interactions may take the form of electrical, chemical, magnetic, thermal or force fields. In many fields of endeavor, the study of the physical properties of an entity and its interaction to these environmental forces has been motivated by a desire to understand the response of an entity to its environment or the desire to interact with it or to create a new and improved version of it.
With the advent of computer technology, those engaged in this type of study were able to leverage an incredibly powerful tool to enhance their ability to understand physical entities and systems. Computational modeling, using the computer to model entities “virtually”, gave rise to many strategies for description and analysis. One such strategy, which began in earnest in the 1950s, came to be widely recognized as having enormous potential for computational modeling, and became known as the Finite Element Method (FEM). In Finite Element Analysis (FEA), a computer model of an entity is used to evaluate its reaction to various forces. The utility of FEA is readily seen in the fact that it is commonly used in nearly every engineering field, and is perhaps the most widely used computational modeling technique used in engineering design and analysis.
When using FEA, a “model” of the entity is created using a computer, and then “virtual” environmental forces are described and applied to the model. Viewing the results provides insight into how a real-world entity would actually behave in a situation similar to the one in the FEA model. Since the 1970s, the ability of the FEM to usefully describe physical phenomena has been widely recognized.
However, one of the limitations of the Finite Element Method is that the properties of the entity must be well understood in order to be modeled accurately. An FEA model can be visualized by thinking of a “Lego® model”, where the “Lego®” pieces are of various sizes and shapes (sometimes very complex shapes). Performing an analysis requires that each “Lego®” piece has a well-defined set of behavior, with respect to the forces the model will use (for example, pressure or heat or electrical forces).
The challenge with FEA modeling is to correctly include in the model the aspects necessary to describe the behavior of interest. Essentially, this correct modeling means sufficiently describing the behavior of each “Lego®” piece in the model, at the level required to evaluate the impact of the “virtual environmental forces” in the problem. This correct modeling has been less of a problem for traditional materials like steel, but has become more of a problem as man-made materials have increased in complexity, especially man-made composites. Additionally, composite materials are common in nature. Examples include wood, bone, the exoskeleton of insects, and many biological tissues. These biological materials are extremely complex, both in behavior and complexity of structure, and have therefore presented difficulties for traditional FEA.
The fundamental difficulty with modeling such materials using FEA stems from the fact that they are entities which have “multiple levels of organization.” For example, consider the hull of a fiberglass boat. In order to model the boat, “Lego®” pieces of the boat hull would need to be in the model. However, fiberglass is a mixture of microscopic glass fibers bonded together with glue, and the orientation and length and strength of the fibers has a dramatic impact on the behavior of a piece of fiberglass. Similarly, bending a piece of fiberglass impacts all the microscopic fibers at the bend, potentially damaging some, and thus impacting the behavior of the piece of fiberglass.
A similar problem exists in many biological tissues because of the inherently hierarchical nature of cells which are organized in various levels to form various types of tissue. Consider skeletal muscle, for example, which is illustrated in FIG. 2. Bulk muscle is composed of a hierarchical structure in which proteins are organized into force-generating units called “Sarcomeres”, which are organized into Myofibrils, which are organized into Muscle Fibers, which are organized into Groups of Muscle Fibers, which compose bulk muscle.
The problem with using the Finite Element Method on a problem like muscle is that accurately modeling the behavior of muscle requires inclusion of the behavior of each of the levels within this hierarchy, as well as the interactions between layers.
Therefore, a need exists for a method to create a coupled system of 2 or more FEA analyses which run concurrently and work together to model an entity with multiple levels of organization.